Finite-element discretizations of a two-dimensional grade-two fluid model
Laboratoire d'Analyse Numérique, Université
Pierre et Marie Curie, 75252 Paris Cedex 05, France.
2 Department of Mathematics, University of Chicago, Chicago, Illinois 60637-1581, USA.
Revised: 3 May 2001
We propose and analyze several finite-element schemes for solving a grade-two fluid model, with a tangential boundary condition, in a two-dimensional polygon. The exact problem is split into a generalized Stokes problem and a transport equation, in such a way that it always has a solution without restriction on the shape of the domain and on the size of the data. The first scheme uses divergence-free discrete velocities and a centered discretization of the transport term, whereas the other schemes use Hood-Taylor discretizations for the velocity and pressure, and either a centered or an upwind discretization of the transport term. One facet of our analysis is that, without restrictions on the data, each scheme has a discrete solution and all discrete solutions converge strongly to solutions of the exact problem. Furthermore, if the domain is convex and the data satisfy certain conditions, each scheme satisfies error inequalities that lead to error estimates.
Mathematics Subject Classification: 65D30 / 65N15 / 65N30
Key words: Mixed formulation / divergence-zero finite elements / inf-sup condition / uniform W1,p-stability / Hood-Taylor method / streamline diffusion.
© EDP Sciences, SMAI, 2001